ON THE MOTION OF PENDULUMS. 81 in an unlimited fluid k = 1, and we see that this value of Jc must be multiplied, in round numbers, by 2, by 3, and by 6^-, in order to account for the observed effect. The value 1*5445 of ttl is so large that the descending series comes into play in the calculation of the function k, while 0*2822 is so small that the ascending series are rapidly convergent. Hence the near agreement of the values of tjp! deduced from the three experiments is a striking confirmation of the theory. The mean of the three is 0*1158, but of course the last figure cannot be trusted. I shall accordingly assume as the value of the square root of the index of friction of air in its average state of pressure, temperature, and moisture vV = 0-116. It is to be remembered that yf/jf expresses a length divided by the square root of a time, and that the numerical value above given is adapted to an English inch as the unit of length, and a second of mean solar time as the unit of time. 56. I now proceed to compare the observed values of tt with those calculated from theory with the assumed value of *Jjju. I begin with the same cylindrical rods as before, together with the long brass tubes Nos. 35 to 38. The diameter of this tube was 1*5 inch, and its length 56 inches. The ends were open, but as the included air was treated by Mr Baily in the reduction of his experiments as if it formed part of the pendulum, we may regard the pendulum as a solid rod. The tube was furnished with six agate planes, represented in the wood-cut at page 417, which rested on fixed knife-edges. The pendulums Nos. 35, 36, 37, and 38 consisted of the same tube swung on the planes marked A, 0, a, c. In air the pendulum swung at the rate of about 90080 vibrations in a day, so that r = 0*9596 nearly. The values of n obtained with the end planes A, c were slightly though sensibly greater than the values obtained with the mean planes (7, a. I shall suppose the mean of the four values of n, namely 2*290, to be the result of the experiments. In the following table the difference between the theoretical and experimental values of tl is exhibited both by decimals and as a fractional part of the former of these values. s. in.